I upgraded from one MBA to another using the Migration Assistant to transfer the account from a Time Capsule. While that worked fine, I cannot access any time machine backups prior to the upgrade--the files are there, but I do not have permissions to view them. It may be that my Time Capsule is logically screwed up (and will need to be reimaged), however these steps allowed me to open the files and I wanted to pass this on to the community.
Using Finder, Navigate to your Time Capsule and open the file for your computer ending in sparsebundle. It will show a folder named backups.backupdb. Open that folder and you will see another folder with your computer name--open that. Then you will see a whole bunch of folders that show dates in the title. Pick one which you believe will contain the files you want. Open it. In this folder you will see the name of your hard drive. Open it. Now you will see folders labeled "applications," "library," "system", "user information", and "users." Open "users." Now you will see a folder for each account and "Shared." Choose the user account of interest.
Now you'll see folders for Desktop, Documents, Downloads, Movies, etc. Each will likely have a red circle with a white line in it (like the international road symbol for "Do Not Enter."). Right click on the one you want to view and select "Get Info". A new window opens.
At the bottom of the new window is a section called "Sharing & Permissions:" Make sure the arrow to the left is pointing down. There is a padlock at the bottom right of the section, click on it. It will ask you for your password (which is the password for your computer). Enter it. There are now three things that can happen:
1. You can select Read & Write for "everyone"
2. You can select the + button at the bottom
3. Both options 1 and 2 are greyed out.
If Option 1 works, change it to Read & Write, click on the Gear at the bottom and select option to the effect of applying these permissions to all subordinate files. Time Machine should now be able to restore from this folder.
If Option 2 works, add your current User name, select Read & Write, click on the Gear at the bottom and select option to the effect of applying these permissions to all subordinate files. Time Machine should now be able to restore from this folder.
If you got to Option 3, you now have to copy this folder somewhere off the Time Capsule. Drag the folder to the desired location. After a few minutes you will be prompted for your computer password. After it copies it will no longer be restricted. You can retrieve the files manually.
Hope someone finds this helpful.